American Institute of Mathematical Sciences

Advanced
September  2007, 6(3): 587-605. doi: 10.3934/cpaa.2007.6.587

On the numerical evaluation of fractional Sobolev norms

Carsten Burstedde 1,

1. 

Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station, C0200, Austin, TX 78712, United States

Received  February 2006 Revised  February 2007 Published  June 2007

In several important and active fields of modern applied mathematics, such as the numerical solution of PDE-constrained control problems or various applications in image processing and data fitting, the evaluation of (integer and real) Sobolev norms constitutes a crucial ingredient. Different approaches exist for varying ranges of smoothness indices and with varying properties concerning exactness, equivalence and the computing time for the numerical evaluation. These can usually be expressed in terms of discrete Riesz operators.
We propose a collection of criteria which allow to compare different constructions. Then we develop a unified approach which is valid for non-negative real smoothness indices for standard finite elements, and for positive and negative real smoothness for biorthogonal wavelet bases. This construction delivers a wider range of exactness than the currently known constructions and is computable in linear time.
Keywords: control problem, finite elements, biorthogonal wavelets., Fractional Sobolev norm, elliptic PDE.
Mathematics Subject Classification: Primary: 65K10, 65J10; Secondary: 65T6.
Citation: Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587
[1]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[2]

Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
[3]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
[4]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems & Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643
[5]

Frédéric Bernicot, Vjekoslav Kovač. Sobolev norm estimates for a class of bilinear multipliers. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1305-1315. doi: 10.3934/cpaa.2014.13.1305
[6]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241
[7]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
[8]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks & Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501
[9]

Eric Dubach, Robert Luce, Jean-Marie Thomas. Pseudo-Conform Polynomial Lagrange Finite Elements on Quadrilaterals and Hexahedra. Communications on Pure & Applied Analysis, 2009, 8 (1) : 237-254. doi: 10.3934/cpaa.2009.8.237
[10]

Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181
[11]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems & Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046
[12]

Irena Pawłow, Wojciech M. Zajączkowski. Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 441-466. doi: 10.3934/dcdss.2011.4.441
[13]

Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041
[14]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266
[15]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[16]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[17]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[18]

Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603
[19]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[20]

Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences